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Properties

Label 2-1805-95.79-c0-0-0
Degree $2$
Conductor $1805$
Sign $0.486 - 0.873i$
Analytic cond. $0.900812$
Root an. cond. $0.949111$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.08 − 0.909i)2-s + (−1.32 + 0.483i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (1.87 + 0.684i)6-s + (0.766 − 0.642i)9-s + (1.08 − 0.909i)10-s + (−0.707 − 1.22i)12-s + (1.32 + 0.483i)13-s + (−0.245 − 1.39i)15-s + (0.939 − 0.342i)16-s − 1.41·18-s − 1.00·20-s + (−0.939 − 0.342i)25-s + (−0.999 − 1.73i)26-s + ⋯
L(s)  = 1  + (−1.08 − 0.909i)2-s + (−1.32 + 0.483i)3-s + (0.173 + 0.984i)4-s + (−0.173 + 0.984i)5-s + (1.87 + 0.684i)6-s + (0.766 − 0.642i)9-s + (1.08 − 0.909i)10-s + (−0.707 − 1.22i)12-s + (1.32 + 0.483i)13-s + (−0.245 − 1.39i)15-s + (0.939 − 0.342i)16-s − 1.41·18-s − 1.00·20-s + (−0.939 − 0.342i)25-s + (−0.999 − 1.73i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.486 - 0.873i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ 0.486 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3357149794\)
\(L(\frac12)\) \(\approx\) \(0.3357149794\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 \)
good2 \( 1 + (1.08 + 0.909i)T + (0.173 + 0.984i)T^{2} \)
3 \( 1 + (1.32 - 0.483i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.32 - 0.483i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
23 \( 1 + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + (-0.245 - 1.39i)T + (-0.939 + 0.342i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (-0.939 + 0.342i)T^{2} \)
67 \( 1 + (1.08 - 0.909i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T^{2} \)
97 \( 1 + (-1.08 - 0.909i)T + (0.173 + 0.984i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.834456604049416608099916883478, −9.115742684208134934491106542606, −8.207416724356486193959675098830, −7.31530723543424235123443459576, −6.21225525464538832219178082596, −5.86880241614770173262660785248, −4.55244719803694623852970634402, −3.59357063301225749018263706735, −2.52610247492520836734286134394, −1.16145643419303792214584905934, 0.54542252911770121957812014451, 1.45888709436266364234402245954, 3.59982608745578708669701935652, 4.80347374695281852946417682883, 5.72703555347456897508434160732, 6.14631190221241046026892179166, 6.95153683815487225207121391669, 7.83839239677226577554220997837, 8.402340590831202336568180776200, 9.117253184464622146965083949953

Graph of the $Z$-function along the critical line